For any simple, undirected graph $G=(V,E)$ where $V$ is finite, we define the Hadwiger number $\eta(G)$ to be the maximum $n$ such that $K_n$ is a minor of $G$.

Is there a graph $G$ on such that removing a vertex or contracting an edge reduces the chromatic number, but has no effect on the Hadwiger number?

To be more precise, I'm looking for a graph $G$ such that

- $G$ is vertex-critical (for any $v\in V(G)$ we have $\chi(G\setminus\{v\}) = \chi(G) - 1$);
- $G$ is edge-contraction-critical (for any $e\in E(G)$ contracting $e$ reduces the chromatic number);
- for any $v\in V(G)$ we have $\eta(G) = \eta(G\setminus \{v\})$;
- for any $e\in E(G)$ contracting $e$ does not change the Hadwiger number.